Abstract: A complex manifold of dimension together with an ample vector bundle on it will be called a generalized polarized variety. The adjoint bundle of the pair is the line bundle . We study the positivity (the nefness or ampleness) of the adjoint bundle in the case . If this was previously done in a series of papers by Ye and Zhang, by Fujita, and by Andreatta, Ballico and Wisniewski.

If is nef then, by the Kawamata-Shokurov base point free theorem, it supports a contraction; i.e. a map from onto a normal projective variety with connected fiber and such that , for some ample line bundle on . We describe those contractions for which . We extend this result to the case in which has log terminal singularities. In particular this gives Mukai's conjecture 1 for singular varieties. We consider also the case in which for every fiber and is birational.